4.1.1. Prof. Ashish Choudhury

Hello everyone, welcome to the first part of tutorial 9 so, let us start with question number 1. In this question you are given 3 positive numbers l, m, n not to be positive, non-negative integers. So, l, m, n such that l is less than equal to m and m is less than equal to n.

The introduction sets the stage for tackling a mathematical problem involving graph theory. Here, 'l', 'm', and 'n' are variables representing certain properties of a graph related to vertex connectivity, edge connectivity, and minimum degree. The problem states specific inequalities that must be respected: l should be less than or equal to m, and m should be less than or equal to n.

Consider a group of friends where some friends are closely connected (like strong friendships) and others are less connected. The numbers 'l', 'm', and 'n' could represent the minimum number of strong connections one needs to form a new friendship group. Just as in the graph, this introductory part explains the limits on how we can 'connect' friendships based on existing connections.

What we want here is a simple graph where the vertex connectivity is l, edge connectivity is m and minimum degree is n. Remember the relationship between the vertex connectivity, edge connectivity, and the minimum degree is that: vertex connectivity is less than equal to edge connectivity and edge connectivity is less than equal to the minimum degree in the graph.

This chunk outlines the criteria for constructing the graph. It emphasizes the relationships among key concepts in graph theory: vertex connectivity (the minimum number of vertices that must be removed to disconnect the graph), edge connectivity (the minimum number of edges that must be removed to disconnect the graph), and minimum degree (the smallest number of edges connected to any single vertex). The requirements help ensure the graph is not just any structure, but meets specific connectivity standards.

Think of a city's road network. The vertex connectivity could represent how many roads must be closed for certain areas to be isolated. Edge connectivity relates to specific roads that connect districts. The minimum degree reflects how many routes each neighborhood has to other parts of the city. Understanding these relationships helps planners ensure reliable connections throughout the city.

Since we need the minimum degree in the graph to be n, I take 2 copies of a complete graph with n + 1 nodes.

The construction of the graph begins with taking two complete graphs containing 'n + 1' nodes each. A complete graph is a type of graph in which every pair of distinct vertices is connected by a unique edge. By using two copies, we ensure a solid foundation that can be manipulated to meet the connectivity requirements expressed by 'l', 'm', and 'n'. This step is critical to achieve the specified minimum degree.

Imagine constructing two tightly knit communities (like clubs), where everyone knows each other. By ensuring everyone in these clubs is connected, we create a strong foundation. This forms the basis for constructing more complex connections as we introduce new relationships later on.

Now I have to take care of my vertex connectivity and edge connectivity. So, how do I do that? I randomly pick l nodes and m nodes from the two copies. So, l nodes I pick from the first copy and m nodes I pick from the second copy.

This step involves selecting specific nodes from the complete graphs to ensure that they fulfill the conditions for vertex and edge connectivity. By picking 'l' nodes from one graph and 'm' from the other, we create the necessary connections that will allow the graph to maintain its required connectivity properties. This allows us to demonstrate flexibility and creativity in how we can build the graph with the original constraints.

It's like selecting team captains from each club to coordinate activities. By having representatives (nodes) from both clubs, we can ensure their collaboration and manage joint events, ensuring that both groups remain well-connected.

I will add extra edges in my graph; those edges will be special edges, and these special edges will ensure that my vertex connectivity of the overall graph is l and edge connectivity of the overall graph is m.

In this phase, additional edges, referred to as 'special edges', are added between the selected nodes from the two graphs to ensure that the graph meets the specified connectivity criteria. These strategically placed edges are key to achieving 'l' and 'm', emphasizing the critical role that edge placement plays in maintaining overall graph connectivity.

Consider adding bridges between two islands that strengthen trade connections. These 'bridges' (special edges) enhance the direct routes between key community representatives (nodes), ensuring that, even if one connection fails, trade can continue through others, thus maintaining connectivity.

If I take any vertex among v1, v2, v3, it is occurring as one of the endpoints out of these 4 edges. And likewise, if I take any of the vertices u1, u2, u3, u4, it is occurring as one of the endpoints of these 4 edges.

This demonstrates that the selected nodes are indeed connected to the special edges added earlier. By ensuring that both the vertices from the first and second complete graph act as endpoints for these added edges, we are guaranteeing that the specified vertex connectivity 'l' is met, which is crucial for the overall integrity of the graph's connectivity.

Imagine that for a sports event, every player from one team (v nodes) is paired with players from another team (u nodes). Their paired relationships ensure that strategies (edges) are formed, ensuring effective teamwork. This creates a network of support throughout the event.

Removing l vertices from the first copy of K graph ensures the entire graph gets disconnected and meets the property that vertex connectivity is l.

This conclusion illustrates how removing certain vertices verifies the designed vertex connectivity of 'l'. If the removal of these nodes leads to disconnection of the graph, it confirms that the conditions set forth in our initial problem are indeed fulfilled.

Think of removing key staff members from a project team. If certain project leads are gone and the team can no longer function properly, it showcases their critical roles in maintaining group effectiveness. Their removal directly affects the project's connectivity.

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